Question

It is desired to make a long cylindrical conductor whose temperature coefficient of resistivity at ${20}^o$C will be close to zero. If such a conductor is made by assembling alternate disks of iron and carbon, find the ratio of the thickness of a carbon disk to that an iron disk. (for carbon, $\rho =3500\times {10}^{-8}\Omega$m and $\alpha =-0.50\times {10}^{-3}$ ${}^o{C}^{-1}$ for iron, $\rho =9.68\times {10}^{-8}\Omega$m and $\alpha =6.5\times {10}^{-3}$ ${}^o{C}^{-1}$).

• A. $0.36$
• B. $0.036$
• C. $1.0$
• D. $2.0$

Answer 1

The correct option is B $0.036$

Change in the resistance of the conductor on increasing its temperature,

$R-R_0=R_0{\alpha }_{av}(T-T_0)$

The disks will be effectively in series, so we will add the resistances to get the total. Looking only at one pair of the disk, we have

$R_c+R_i=R0c\left({\alpha }_c\right(T-T_0)+1)+R_{0i}\left({\alpha }_i\right(T-T_0)+1)$

$=R_{0c}+R_{0i}+(R_{0c}{\alpha }_c+R_{0i}{\alpha }_i)(T-T_0)$

This equation will only be constant if the coefficient for the term $(T-T_0)$ vanishes.

Then $R_{0c}{\alpha }_c+R_{0i}{\alpha }_i=0$

but $R=\frac{\rho L}{A}$, and the disks have the same cross sectional area, so, $L_c{\rho }_c{\alpha }_c+L_i{\rho }_i{\alpha }_i=0$

or, $\frac{L_c}{L_i}=\frac{{\rho }_i{\alpha }_i}{{\rho }_c{\alpha }_c}=-\frac{9.68\times {10}^{-8}\times 6.5\times {10}^{-3}}{3500\times {10}^{-8}\times (-0.50\times {10}^{-3})}=0.036$

Hence option $\left(B\right)$ is correct.